<Tri-2024-2116>, hint

Let A(-a,o), B(0,b) , a>b>0 O(0,0) <BAO=@, tan@=b/a P(-p,0,) 0<p<a 

Slope of PQ=tan(@+45)=(a+b)/(a-b)

Since OPQ is right triangle and M is the midpoint of PQ, MO=MA=MB

MPO is isosceles triangle  , Hence  OM : y=-(a+b)/(a-b) x

PQ : y=(a+b)/(a-b)(x+p), Hence Q(0, p(a+b)/(a-b))

MKL : (y-p(a+b)/2(a-b)=-(a-b)/(a+b)(x-p/2) x=0 L(0, p/2 {(a+b)/(a-b)-(a-b)/(a+b)}

Triangles OPQ and OLK are similar Since OL>OP, LK>PQ